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Once you have the strange amulet, go and talk to the wizard about it instead of talking to the steward and accusing him. The wizard will tell you when the killer will strike next, and you can "catch" him in the act (and by "catch" I mean "fill with arrows").

The court wizard can usually be found in his quarters in the Jarl's residence, his room is the only one with an arcane enchanter.

Unfortunately, you can't skip ahead just by pickpocketing the guy and looking at his journal - after all, it's not that incriminating, he could be writing a story or something.